1. Technical Field
This application relates to the field of a dynamically matched set of golf clubs.
2. Background of Invention
FIG. 1 shows the typical golf club comprising a shaft 30, a grip 32 located at one end of the shaft referred to as the grip or butt end 34, and a clubhead 36 located at the other end of shaft 30. The forward end of clubhead 36 is known as the toe 38 and the opposite end is known as the heel 40.
The matching of golf clubs in the past has involved both static and dynamic methods. The static method is often referred to as the swing-weight method which correlates the relative weight in the club head to the overall weight. By definition (U.S. Pat. Nos. 1,953,916 and 1,594,801) the swing-weight is the moment of the club's weight about a fulcrum 12 inches (30.48 cm) from the grip end of the club. The value of the moment is correlated to an index consisting of a letter and a number, C2, C3, . . . C9, D0, D1 . . . D9 . . . , representing increasing moment. That is, the moment generated by a D2 club is greater than the moment generated by a C2 club. A set of clubs is matched by making every club in the set a specific swing-weight. For example, a set of clubs may be matched by making all the clubs have a swing-weight of D0.
Dynamic correlation of golf clubs is a second way of matching in which the moment of inertia, I, or the radius of gyration, k, is prescribed for a set of clubs. Prescribing the moment of inertia about a given axis is tantamount to prescribing the radius of gyration about that axis because the two are related through the mass of the club, m, as follows: EQU k.sup.2 I/m. (1)
The value of the moment of inertia, and hence the radius of gyration, is dependent on the axis of rotation.
U.S. Pat. Nos. 4,128,242; 5,094,101; and 3,698,239 claim to match the moment of inertia of a set of golf clubs as they rotate about a single axis which may correspond to the grip end of the club, the center of gravity, or the center of percussion. The deficiency in dynamically matching clubs with these methods is that only one axis, and one corresponding moment of inertia is considered. This is not sufficient to fully characterize the dynamic response of the club since there are two remaining axes to measure the moments of inertia as well as the products of inertia since, in general, the axis of interest may not be principal axes.
In order to dynamically characterize a golf club, in accordance with this invention, one must know the mass, m, the center of gravity, G, and the inertia tensor ##EQU1## where I.sub.xx, I.sub.yy, and I.sub.zz are moments of inertia about the x, y, and z axes, respectively, and I.sub.xy, I.sub.xz, and I.sub.yz are the products of inertia.
As a simple example, consider the right-cylinder of FIG. 2 as an example of the ienrtia tensor. An dynamics book (Beer and Johnson, 1988) gives the inertia tensor with respect to the centroidal, or center of gravity, axes as follows: ##EQU2## where m is the mass of the cylinder. The products of ienrtia are zero because of the symmetry of the material with respect to axes &lt;xyz&gt;. The inertia tensor with respect to axes &lt;x.sub.1 y.sub.1 z.sub.1 &gt; can be easily determined to be ##EQU3## In order to fully describe the dynamic response of a club, one must determine the six inertia terms I.sub.xx, I.sub.yy, I.sub.zz, I.sub.xy, I.sub.xz, and I.sub.yz referred to a specific reference frame, the mass, and the center of gravity.
Once this is specified, the ienrtia tensor can be described for any other axis parallel to the centroidal axes by using the parallel axis therorem (Beer and Johnson, 1988), ##EQU4## where &lt;xyz&gt; and &lt;x.sub.1 y.sub.1 z.sub.1 &gt; are centroidal axes and an arbitrary set of axes parallel to the centroidal axes, respectively. In the second term of each of the six equations, x, y, and z are the perpendicular distance that x is from x.sub.1, y is from y.sub.1, and z is from z.sub.I, respectively.
If the inertia tensor components are desired at an axis, &lt;x.sub.2 y.sub.2 z.sub.2 &gt;, which is neither at the centroid nor has all three axes parallel to the centroidal axes, &lt;xyz&gt;, then the following transformation must be applied to the inertia tensor after the parallel axis theorem, Eq. 5, has been applied: ##EQU5## where EQU .alpha..sub.11 =cos(x.sub.2,x.sub.1),.alpha..sub.12 =cos(x.sub.2,y.sub.1),.alpha..sub.13 =cos(x.sub.2,z.sub.1) EQU .alpha..sub.21 =cos(y.sub.2,x.sub.1),.alpha..sub.31 =cos(z.sub.2,x.sub.1),.alpha..sub.32 =cos(z.sub.2,y.sub.1)7)
(Ref. Mase and Mase, 1992).